Seminar

An overview of oscillatory integrals and integral operators in high frequency scattering

Chandler-Wilde, S (Reading)Wednesday 17 January 2007, 14:30-15:30

Seminar Room 1, Newton Institute

Abstract

In this talk we review recent research on integral equation methods for high frequency time-harmonic scattering. A number of interesting questions arise in this context, including:

* The kernels of the boundary integral operators become increasingly oscillatory as the frequency increases. What is the dependence on frequency of the condition numbers of these operators? What can be said about this after discretisation?

* Is it possible, by use of an ansatz based on high frequency asymptotics, to represent oscillatory solutions with a number of degrees of freedom which is fixed as the frequency increases?

* Using such an ansatz, the entries of an approximating linear system are oscillatory integrals. What form do these integrals take and how can we evaluate them?

* For which classes of problems might we hope to achieve computational cost which is O(1) in its dependence on the frequency as the frequency increases?

We will indicate the (exciting but limited) progress that has been made internationally in addressing these questions in the last few years, and give some idea of the methods that have been used, focussing particularly on the simplest problem of scattering of time-harmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions. We will also list some of the many open, difficult problems in this area.